Images of Julia sets that you can trust
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1 Images of Julia sets that you can trust Luiz Henrique de Figueiredo with Diego Nehab (IMPA) Jorge Stolfi (UNICAMP) João Batista Oliveira (PUCRS)
2 Can we trust this beautiful image? Curtis McMullen
3 Julia sets Study the dynamics of f (z) = z 2 + c for c C fixed z 1 = f (z 0 ), z 2 = f (z 1 ),..., z n = f (z n 1 ) = f n (z 0 ) What happens with the orbit of z 0 C under f?
4 Julia sets M M unbounded orbits bounded orbits
5 Julia sets M unbounded orbits attraction basin of A( ) M bounded orbits
6 Julia sets M unbounded orbits attraction basin of A( ) M bounded orbits filled Julia set K
7 Julia sets M unbounded orbits attraction basin of A( ) M bounded orbits filled Julia set K M common boundary Julia set J
8 Julia set zoo Glenn Elert
9 Julia set catalog: the Mandelbrot set c M := 0 K c Paul Bourke Julia Fatou dichotomy c M J c is connected c M J c is a Cantor set
10 Julia set catalog: the Mandelbrot set demo... c M := 0 K c Paul Bourke Julia Fatou dichotomy c M J c is connected c M J c is a Cantor set
11 Why distrust this beautiful image? Curtis McMullen
12 Why distrust this beautiful image? Escape-time algorithm for z 0 in a grid of points in Ω z z 0 n 0 while z R and n N do z z 2 + c n n + 1 paint z 0 with color n Curtis McMullen
13 Why distrust this beautiful image? Escape-time algorithm for z 0 in a grid of points in Ω z z 0 n 0 while z R and n N do z z 2 + c n n + 1 paint z 0 with color n Curtis McMullen escape radius R = max( c, 2) J B(0, R)
14 Escape radius Lemma. If z C and z > R = max( c, 2) f n (z) as n. Proof. The triangle inequality gives z 2 = z 2 + c c z 2 + c + c and so f (z) = z 2 +c z 2 c = z 2 c > z 2 z = z ( z 1) > z > R Iterating, we get f n (z) > z ( z 1) n because z 1 > 1. Corollary. Every unbounded orbit escapes to. A( )
15 Why distrust this beautiful image? Escape-time algorithm for z 0 in a grid of points in Ω z z 0 n 0 while z R and n N do z z 2 + c n n + 1 paint z 0 with color n Curtis McMullen escape radius R = max( c, 2) J B(0, R)
16 Why distrust this beautiful image? Escape-time algorithm for z 0 in a grid of points in Ω z z 0 n 0 while z R and n N do z z 2 + c n n + 1 paint z 0 with color n Curtis McMullen
17 Why distrust this beautiful image? Escape-time algorithm Spatial sampling need fine grid what happens between samples? for z 0 in a grid of points in Ω z z 0 n 0 while z R and n N do z z 2 + c n n + 1 paint z 0 with color n
18 Why distrust this beautiful image? Escape-time algorithm Spatial sampling Partial orbits program cannot run forever for z 0 in a grid of points in Ω z z 0 n 0 while z R and n N do z z 2 + c n n + 1 paint z 0 with color n
19 Why distrust this beautiful image? Escape-time algorithm Spatial sampling Partial orbits for z 0 in a grid of points in Ω z z 0 n 0 while z R and n N do z z 2 + c n n + 1 paint z 0 with color n Floating-point rounding errors squaring needs double digits
20 Why distrust this beautiful image? Spatial sampling Compute color on grid points Cannot be sure grid is fine enough Cannot be sure behavior at sample points is typical Finer grid more detail Partial orbits Can only compute partial orbits Cannot be sure partial orbits are long enough Longer orbits more detail Floating-point errors z 2 needs twice the number of digits that z needs Do rounding errors during iteration influence classification of points? Multiple-precision more detail (deep zoom)
21 You can trust our method No spatial sampling No orbits No floating-point errors
22 You can trust our method No spatial sampling Classify entire rectangles in the complex plane Spatial resolution limited by available memory Deeper quadtree more detail No orbits No floating-point errors
23 You can trust our method No spatial sampling Classify entire rectangles in the complex plane Spatial resolution limited by available memory Deeper quadtree more detail No orbits Evaluate f once on each cell using interval arithmetic Perform no function iteration at all Use cell mapping and color propagation in graphs No floating-point errors
24 You can trust our method No spatial sampling Classify entire rectangles in the complex plane Spatial resolution limited by available memory Deeper quadtree more detail No orbits Evaluate f once on each cell using interval arithmetic Perform no function iteration at all Use cell mapping and color propagation in graphs No floating-point errors All numbers are dyadic fractions with restricted range and precision Use error-free fixed-point arithmetic Precision depends only on spatial resolution Standard double-precision floating-point enough for huge images
25 Our algorithm quadtree for Ω = [ R, R] [ R, R] white rectangles contained in A( ) black rectangles contained in K gray rectangles contain J
26 Our algorithm quadtree for Ω = [ R, R] [ R, R] white rectangles contained in A( ) black rectangles contained in K gray rectangles contain J certified decomposition
27 Our algorithm quadtree for Ω = [ R, R] [ R, R] refinement cell mapping color propagation
28 Our algorithm quadtree for Ω = [ R, R] [ R, R] refinement cell mapping color propagation
29 Quadtree c = 1 level 0
30 Quadtree c = 1 level 1
31 Quadtree c = 1 level 2
32 Quadtree c = 1 level 3
33 Quadtree c = 1 level 4
34 Quadtree c = 1 level 5
35 Quadtree c = 1 level 6
36 Quadtree c = 1 level 7
37 Quadtree c = 1 level 8
38 Quadtree c = 1 level 9
39 Quadtree c = 1 level 10
40 Quadtree c = 1 level 11
41 Quadtree c = 1 level 12
42 Quadtree c = 1 level 13
43 Quadtree c = 1 level 14
44 Adaptive approximation c = 1 level 14
45 Adaptive approximation c = 1 level 0
46 Adaptive approximation c = 1 level 1
47 Adaptive approximation c = 1 level 2
48 Adaptive approximation c = 1 level 3
49 Adaptive approximation c = 1 level 4
50 Adaptive approximation c = 1 level 5
51 Adaptive approximation c = 1 level 6
52 Adaptive approximation c = 1 level 7
53 Adaptive approximation c = 1 level 8
54 Adaptive approximation c = 1 level 9
55 Adaptive approximation c = 1 level 10
56 Adaptive approximation c = 1 level 11
57 Adaptive approximation c = 1 level 12
58 Adaptive approximation c = 1 level 13
59 Adaptive approximation c = 1 level 14
60 Adaptive approximation c = 1
61 Adaptive approximation c = 1
62 Our algorithm quadtree for Ω = [ R, R] [ R, R] refinement cell mapping color propagation
63 Cell mapping Directed graph on the leaves of the quadtree and exterior edges emanate from each leaf gray cell A add edge A B for each leaf cell B that intersects f (A) f (A) B A B
64 Cell mapping Directed graph on the leaves of the quadtree and exterior edges emanate from each leaf gray cell A add edge A B for each leaf cell B that intersects f (A) f (A) B A B Conservative estimate of the dynamics Avoid point sampling
65 Cell mapping source cell leaf gray cell
66 Cell mapping exact image under f
67 Cell mapping bounding box interval arithmetic
68 Cell mapping quadtree traversal
69 Cell mapping target cells contain exact image
70 Cell mapping edges
71 Cell mapping edges demo...
72 Our algorithm quadtree for Ω = [ R, R] [ R, R] refinement cell mapping color propagation
73 Color propagation Propagate white and black to gray cells new white cells gray cells for which all paths end in white cells new black cells gray cells for which no path ends in a white cell
74 Color propagation Propagate white and black to gray cells new white cells gray cells for which all paths end in white cells new black cells gray cells for which no path ends in a white cell Graph traversals replace function iteration Avoid floating-point errors
75 The algorithm initial approximation
76 The algorithm refinement
77 The algorithm cell mapping
78 The algorithm new white cells...
79 The algorithm new white cells...
80 The algorithm new white cells
81 The algorithm gray cells that reach white...
82 The algorithm gray cells that reach white...
83 The algorithm gray cells that reach white
84 The algorithm new black cells
85 Adaptive approximation examples
86 Adaptive approximation c = i level 0
87 Adaptive approximation c = i level 1
88 Adaptive approximation c = i level 2
89 Adaptive approximation c = i level 3
90 Adaptive approximation c = i level 4
91 Adaptive approximation c = i level 5
92 Adaptive approximation c = i level 6
93 Adaptive approximation c = i level 7
94 Adaptive approximation c = i level 8
95 Adaptive approximation c = i level 9
96 Adaptive approximation c = i level 10
97 Adaptive approximation c = i level 11
98 Adaptive approximation c = i level 12
99 Adaptive approximation c = i level 13
100 Adaptive approximation c = i level 14
101 Adaptive approximation c = i
102 Adaptive approximation c = i
103 Adaptive approximation c = i level 0
104 Adaptive approximation c = i level 1
105 Adaptive approximation c = i level 2
106 Adaptive approximation c = i level 3
107 Adaptive approximation c = i level 4
108 Adaptive approximation c = i level 5
109 Adaptive approximation c = i level 6
110 Adaptive approximation c = i level 7
111 Adaptive approximation c = i level 8
112 Adaptive approximation c = i level 9
113 Adaptive approximation c = i level 10
114 Adaptive approximation c = i level 11
115 Adaptive approximation c = i level 12
116 Adaptive approximation c = i level 13
117 Adaptive approximation c = i level 14
118 Adaptive approximation c = i
119 Adaptive approximation c = i
120 Adaptive approximation c = i level 0
121 Adaptive approximation c = i level 1
122 Adaptive approximation c = i level 2
123 Adaptive approximation c = i level 3
124 Adaptive approximation c = i level 4
125 Adaptive approximation c = i level 5
126 Adaptive approximation c = i level 6
127 Adaptive approximation c = i level 7
128 Adaptive approximation c = i level 8
129 Adaptive approximation c = i level 9
130 Adaptive approximation c = i level 10
131 Adaptive approximation c = i level 11
132 Adaptive approximation c = i level 12
133 Adaptive approximation c = i level 13
134 Adaptive approximation c = i level 14
135 Adaptive approximation c = i
136 Adaptive approximation c = i
137 Adaptive approximation c = i level 0
138 Adaptive approximation c = i level 1
139 Adaptive approximation c = i level 2
140 Adaptive approximation c = i level 3
141 Adaptive approximation c = i level 4
142 Adaptive approximation c = i level 5
143 Adaptive approximation c = i level 6
144 Adaptive approximation c = i level 7
145 Adaptive approximation c = i level 8
146 Adaptive approximation c = i level 9
147 Adaptive approximation c = i level 10
148 Adaptive approximation c = i level 11
149 Adaptive approximation c = i level 12
150 Adaptive approximation c = i level 13
151 Adaptive approximation c = i level 14
152 Adaptive approximation c = i
153 Adaptive approximation c = i
154 Adaptive approximation c = i level 0
155 Adaptive approximation c = i level 1
156 Adaptive approximation c = i level 2
157 Adaptive approximation c = i level 3
158 Adaptive approximation c = i level 4
159 Adaptive approximation c = i level 5
160 Adaptive approximation c = i level 6
161 Adaptive approximation c = i level 7
162 Adaptive approximation c = i level 8
163 Adaptive approximation c = i level 9
164 Adaptive approximation c = i level 10
165 Adaptive approximation c = i level 11
166 Adaptive approximation c = i level 12
167 Adaptive approximation c = i level 13
168 Adaptive approximation c = i level 14
169 Adaptive approximation c = i
170 Adaptive approximation c = i
171 Applications Image generation Point and box classification Fractal dimension of Julia set Area of filled Julia set Diameter of Julia set
172 Applications certified numerical results Image generation large images smaller images with anti-aliasing Point and box classification quadtree traversal + one function evaluation if gray Fractal dimension of Julia set (Ruelle) upper bound dim H = 1 + c 2 4 log 2 + Area of filled Julia set (Milnor) lower and upper bounds π(1 p 1(c) 2 3 p 2(c) 2 5 p 3(c) 2 ) Diameter of Julia set lower and upper bounds
173 Area of filled Julia sets after Milnor Inverse Böttcher map ψ : C \ D C \ K Laurent series near a 2 = c 2 ψ(w) = w Gronwall s area theorem ψ(w 2 ) = ψ(w) 2 + c ( 1 + a 2 w 2 + a 4 w 4 + a 6 w 6 + ) a 2n = 1 2 (a n an) 2 a j a 2n j a 2n+1 = 0 2 j<n j even area(k) = π(1 a a a 6 2 ) Truncating series gives upper bounds Quadtree gives both lower and upper bounds series converges slowly
174 rees with the (l/r)-metric Idzl/r. Area of filled Julia sets after Milnor Figure 45. Upper bounds for the area of the filled Julia set for fc(z) = z2 + c in the range -2::; c :::;.25. J. Milnor, Dynamics in one complex variable
175 Area of filled Julia set 1.25 c 0.25
176 Area of filled Julia set 1.25 c 0.25 level upper bound lower bound
177 Area of filled Julia set 1.25 c 0.25 level milnor upper bound lower bound
178 Area of filled Julia set 1.25 c 0.25 level error
179 Limitations Memory Need to explore Ω [ R, R] [ R, R]
180 Limitations Memory depth of quadtree and size of cell graph limited by available memory currently spatial resolution cannot reach 20 levels Need to explore Ω [ R, R] [ R, R] even if region of interest is smaller limited amount of zoom limitation inherent to using cell mapping because f is transitive on J
181 Future work higher-degree polynomials Escape radius Bounding box
182 Future work higher-degree polynomials Escape radius R = 1 + a d + + a 0 a d is an escape radius for f (z) = a d z d + + a 0 (Douady) Bounding box needs interval arithmetic with directed rounding
183 Cubic Julia set z
184 Cubic Julia set z
185 Cubic Julia set z
186 Cubic Julia set z 3 3a 2 + b level 0
187 Cubic Julia set z 3 3a 2 + b level 1
188 Cubic Julia set z 3 3a 2 + b level 2
189 Cubic Julia set z 3 3a 2 + b level 3
190 Cubic Julia set z 3 3a 2 + b level 4
191 Cubic Julia set z 3 3a 2 + b level 5
192 Cubic Julia set z 3 3a 2 + b level 6
193 Cubic Julia set z 3 3a 2 + b level 7
194 Cubic Julia set z 3 3a 2 + b level 8
195 Cubic Julia set z 3 3a 2 + b level 9
196 Cubic Julia set z 3 3a 2 + b level 10
197 Cubic Julia set z 3 3a 2 + b level 11
198 Cubic Julia set z 3 3a 2 + b level 12
199 Cubic Julia set z 3 3a 2 + b level 13
200 Cubic Julia set z 3 3a 2 + b level 14
201 Cubic Julia set z 3 3a 2 + b
202 Cubic Julia set z 3 3a 2 + b
203 Future work Newton s method Which points converge to which root? Cayley (1879) Points that do not converge form the Julia set No escape radius Need to find explicit attracting regions around roots?
204 Future work Newton s method z 3 = 1 Joachim von zur Gathen and Jürgen Gerhard
205 Julia set panorama img=../julia-256gp/julia.xml
206 Images of Julia sets that you can trust Thanks!
207 Related work M. Braverman and M. Yampolsky. Computability of Julia sets, volume 23 of Algorithms and Computation in Mathematics. Springer-Verlag, M. Dellnitz and A. Hohmann. A subdivision algorithm for the computation of unstable manifolds and global attractors. Numerische Mathematik, 75(3): , C. S. Hsu. Cell-to-cell mapping: A method of global analysis for nonlinear systems. Springer-Verlag, J. Milnor. Dynamics in one complex variable, volume 160 of Annals of Mathematics Studies. Princeton University Press, third edition, R. E. Moore. Interval Analysis. Prentice-Hall, R. Rettinger and K. Weihrauch. The computational complexity of some Julia sets. In Proceedings of the 35th Annual ACM Symposium on Theory of Computing, pages ACM, D. Saupe. Efficient computation of Julia sets and their fractal dimension. Phys. D, 28(3): , 1987.
208 Area of filled Julia set 1.25 c 0.25 level time
209 Interval arithmetic f (z) = z 2 + c (x, y) (x 2 y 2 + a, 2xy + b) function f(xmin,xmax,ymin,ymax) local x2min,x2max=isqr(xmin,xmax) local y2min,y2max=isqr(ymin,ymax) local xymin,xymax=imul(xmin,xmax,ymin,ymax) return x2min-y2max+a,x2max-y2min+a,2*xymin+b,2*xymax+b end function imul(xmin,xmax,ymin,ymax) local mm=xmin*ymin local mm=xmin*ymax local Mm=xmax*ymin local MM=xmax*ymax local m,m=mm,mm if m>mm then m=mm elseif M<mM then M=mM end if m>mm then m=mm elseif M<Mm then M=Mm end if m>mm then m=mm elseif M<MM then M=MM end return m,m end
210 Interval arithmetic f (z) = z 2 + c (x, y) (x 2 y 2 + a, 2xy + b) function f(xmin,xmax,ymin,ymax) local x2min,x2max=isqr(xmin,xmax) local y2min,y2max=isqr(ymin,ymax) local xymin,xymax=imul(xmin,xmax,ymin,ymax) return x2min-y2max+a,x2max-y2min+a,2*xymin+b,2*xymax+b end function isqr(xmin,xmax) local u=xmin^2 local v=xmax^2 if xmin<=0 and 0<=xmax then if u<v then return 0,v else return 0,u end else if u<v then return u,v else return v,u end end end
211 Interval arithmetic f (z) = z 3 + c (x, y) (x 3 3xy 2 + a, y 3 + 3x 2 y + b) function f(xmin,xmax,ymin,ymax) local x2min,x2max=isqr(xmin,xmax) local y2min,y2max=isqr(ymin,ymax) local xy2min,xy2max=imul(xmin,xmax,y2min,y2max) local x2ymin,x2ymax=imul(x2min,x2max,ymin,ymax) local x3min,x3max=icub(xmin,xmax) local y3min,y3max=icub(ymin,ymax) return x3min-3*xy2max+a, x3max-3*xy2min+a, -y3max+3*x2ymin+b,-y3min+3*x2ymax+b end function icub(xmin,xmax) return xmin^3,xmax^3 end
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